A. Stern's analysis of the nodal sets of some families of spherical harmonics revisited

TL;DR

This paper revisits classical analyses of spherical harmonics with specific nodal domain structures, providing sharper quantitative insights and understanding of bifurcations in these families.

Contribution

It introduces a new method that refines previous results on spherical harmonics, especially regarding their nodal sets and bifurcation phenomena.

Findings

Sharp quantitative results on nodal sets

Enhanced understanding of bifurcation occurrences

Extension of classical analyses with modern techniques

Abstract

In this paper, we revisit the analyses of Antonie Stern (1925) and Hans Lewy (1977) devoted to the construction of spherical harmonics with two or three nodal domains. Our method yields sharp quantitative results and a better understanding of the occurrence of bifurcations in the families of nodal sets.This paper is a natural continuation of our critical reading of A. Stern's results for Dirichlet eigenfunctions in the square, see arXiv:14026054.